1.1 Textbook Learning Objectives

• Interpret the chi-square probability distribution as the sample size changes.
• Conduct and interpret chi-square goodness-of-fit hypothesis tests.
• Conduct and interpret chi-square test of independence hypothesis tests.
• Conduct and interpret chi-square homogeneity hypothesis tests.

1.2 Instructor Learning Objectives

• Appreciate how the chi-square family of tests largely depart from the previous mean-based testing in t-tests
• Understand what scenarios each form of the chi-square test may be appropriate to
• See how the chi-square family of tests add to our toolbox of inferential tests

1.3 Introduction

• The \(\chi^2\) distribution, also known as chi-square distribution, or chi-squared distribution is a unique tool that can be applied in inferential testing
  • Much like the t-distribution, the chi-square distribution serves a purpose of comparison
• There are 3 primary uses for the chi-square distribution:
  • Goodness-of-fit: to compare if data is distributed as expected
  • Test of independence: to compare if two events are independent
  • Test of homogeneity: to test if two distribution are the same as one another

• Important: Though not explicitly mentioned in the book, the chi-square distribution is particularly well-suited to helping us make sense of categorical data - but that is not the ONLY way it can be used

2.1 Formula & Notation

• Discuss: Review: Write out the notation for Student's t-distribution

• Formula for \(\chi^2\) distribution:
  • \(X \sim \chi^2_{df}\)
  • Where:
    • \(df\): degrees of freedom
    • \(X\): a random variable (much like other distributions, this can really be any uppercase letter, X is just convention)
    • \(\chi^2\): arbitrary notation for the chi-square distribution

2.2 Distribution Characteristics

• Discuss: Review: Think back to our initial discussions on the uniform and exponential distributions - write out the formulas for the find long-term mean and standard deviation in those distributions?

• Characteristics of the distribution:
  • \(\mu = df\)
  • \(\sigma = \sqrt{2(df)}\)
  • The curve is nonsymmetrical and skewed to the right
  • The chi-square curve for each \(df\)
  • The test statistic for any test is always greater than or equal to 0
  • When \(df > 90\), the chi-square distribution is approximately normal
  • The \(\mu\) is located to the near right of the “peak” of the curve

• The \(\chi^2\) distribution thus changes dependent on the degrees of freedom

2.3 The df for Chi-square Distribution

• So how does one calculate the important \(df\) for this distribution?
  • Depends on which of the 3 use cases we are implementing

• Discuss: Try explaining, in your own words, what degrees of freedom are again?

• Important: The fact that the chi-square distribution changes dependent on the df is a unique change compared to the normal or t-distributions!

3.1 Introduction

• The goodness-of-fit test is an application where the chi-square distribution is used to determine whether data “fits” a certain distribution

• Discuss: Why might we want to know if the data we have fit a certain distribution? hint: think about 'assumptions'

3.2 Formula

\[ \chi^2 = \sum_k\frac{(O - E)^2}{E} \]

\[ df = k - 1 \]

• Where:
  • O = observed values are those that we actually measure from our sample
  • E = expected values are those that we expect our observed values to be if they follow the same distribution, i.e., if the null hypothesis is true
  • \(k\) = the number of different categories, you can think of this as the “pairs” of observed and expected data points

• Discuss: A chi-square statistic characteristic is always equal to or greater than 0; explain why this characteristic is true by looking at the formula

• Important: Due to some mathematical oddities, don't use this unless expected values are at least 5 or greater in each category - can inflate type I error

3.3 Null and Alternative Hypotheses Testing

• Important: Null and alternative hypotheses can and should be stated when doing any and all inferential testing! Still applies here

• Hypothesis pair:
  • Null hypothesis: The observed values/proportions are the same as the expected values
  • Alternative hypothesis: The observed values/proportions are different than the expected values
• Goodness-fit-tests are usually done in right-tailed manner, as a greater statistic suggest a greater divergence from expected distribution

• Discuss: Draw a normal curve and shade it in such a manner to show a right-tailed test

3.4 Application Example

• I have a class of 20 students, who I expect will have a uniform distribution of As, Bs, Cs, Ds, and Fs
  • \(E: \{4, 4, 4, 4, 4\}\)
  • \(O: \{2, 4, 12, 1, 1\}\)
• Hypothesis pair:
  • \(H_0:\) the observed letter grades fits the expected distribution of 4 of each A, B, C, D, F
  • \(H_A:\) the observed letter grades do not fit the expected distribution of 4 of each A, B, C, D, F

• \(\chi^2 = 21.50\), \(df = 5 - 1 = 4\)
  • \(p\) to be determined by computer and compared against arbitrary \(\alpha\)

• Discuss: Do the same process as demonstrated above, but now using 1,1,1,2,15 as the observed data

4.1 Introduction

• The test of independence test is used when we want to determine if two factors or variables are independent or not
  • This test will use a contingency table that shows frequencies of two variable in data

• Discuss: Think back to our probability units, what did 'independent' mean in that context and how did it relate to sampling?

• Important: Much like goodness-of-fit, don't use this unless expected values are at least 5 or greater in each category - can inflate type I error

4.2 Formula

\[ \chi^2 = \sum_{i * j}\frac{(O - E)^2}{E} \]

\[ df = (i - 1) * (j - 1) \]

• Where:
  • \(O:\) observed values
  • \(E:\) expected values*
  • \(i:\) number of rows in the data
  • \(j:\) number of columns in the data
• Expected value (\(E\)) for each cell of the table is calculated as:
  • \(E = \frac{\sum_{row} * \sum_{column}}{{\sum_{total}}}\)

4.3 Null and Alternative Hypotheses Testing

• Hypothesis pair:
  • Null hypothesis: The two factors are independent
  • Alternative hypothesis: The two factors are dependent

4.4 Application Example

• I have a class of 40 students, half of whom are female and half are male, and below are their counts of of As, Bs, Cs, Ds, and Fs
  • \(F: \{6, 5, 3, 5, 1\}\)
  • \(M: \{3, 4, 4, 7, 2\}\)
• Hypothesis pair:
  • \(H_0:\) The two factors of sex and grade are independent
  • \(H_A:\) The two factors of sex and grade are dependent
• Contingency Table:

• Expected Counts:

• Chi-square statistic:

• \(df = (5 - 1)(2 - 1) = 4\), \(p\) to be calculated by computer

• Discuss: Do the same process as demonstrated above, but now using F: 1,1,1,2,15 and M: 2,2,2,10,4 as the observed data

5.1 Introduction

• Whereas the goodness-of-fit test can tell us whether data matches a specified, known distribution, we have a different test for testing whether two different sets of data follow the same, unknown distribution
  • We call these are test for homogeneity

• Discuss: In what hypothesis testing scenarios might we want to know whether two sets of data seem to have equal distributions? hint: think independent-samples t-test

5.2 Formula

• Important: The chi-square statistic here is calculated the same as with the test of independence, but the df is different!

\[ \chi^2 = \sum_{i * j}\frac{(O - E)^2}{E} \]

\[ df = (i - 1) * (j - 1) \]

• Where:
  • \(O:\) observed values
  • \(E:\) expected values*
  • \(i:\) number of rows in the data
  • \(j:\) number of columns in the data
  • The columns in this setup should be the categorical variable whose distribution we are interested in
  • The rows should be the grouping categorical variable, or the two groups we compare against one another

5.3 Null and Alternative Hypotheses Testing

• Hypothesis pair:
  • Null hypothesis: The distributions of the two populations are the same
  • Alternative hypothesis: The distributions of the two populations are not the same

5.4 Application Example

• I have a group of students, half of whom are female and half are male, and below are their counts of what class they are in:
  • Freshman, Sophomore, Junior, Senior
  • \(F: \{6, 14, 8, 12\}\)
  • \(M: \{5, 7, 7, 8\}\)
• Hypothesis pair:
  • \(H_0:\) The distributions of class between male and female students are the same
  • \(H_A:\) The distributions of class between male and female students are not the same
• Data setup

• Expected counts

• Chi-square calculation

6.1 Recap

• The chi-squared distribution is well suited to dealing with categorical counts and proportions, but the 3 forms of the chi-square distribution have somewhat different application and formulas

• The shape of the chi-square distribution changes based upon the dfs, and the hypothesis testing done with it is usually right-tailed, among other consistent characteristics

• Like with the other distributions we have discussed, be mindful of how the null and alternative hypotheses change

6.2 Lecture Check-in

• Make sure to complete any lecture check-in tasks associated with this lecture!

Module 11 Lecture - Review of Prior Topics || Analysis of Variance